Summary

In Chapter 2, we define the quantum oracle model, and
present Ambainis' Theorem for proving lower bounds within this
framework. We examine a shortcoming of Ambainis' Theorem, and then
derive two theorems and use them to prove lower query bounds for
symmetric functions.

In Chapter 3, we apply the results of Chapter
2 to the interesting and well studied case of
non-trivial monotone graph properties. Finding the results wanting we
appeal directly to Ambainis' Theorem to establish new lower bounds for
graph connectivity and bipartiteness. Finally we show there is no
quantum extension of the Aanderaa-Karp-Rosenberg conjecture in the
bounded error setting.

In Chapter 4, we examine classes of functions that do
not have the inherent symmetries of those in chapters 2
and 3. We provide lower query bounds for tree functions,
nondeterministically evasive functions, and sensitive functions.

We conclude in Chapter 5 with a brief examination of some
open questions for lower oracle query bounds in the bounded error
setting, and quantum computing in general.

Many of the results in this thesis have been attained previously by
other authors. We present these results again to illustrate how
Ambainis' Theorem can provide relatively simpler proofs for a variety
of problems. Table 1.1 summarizes our results and
credits previous papers where appropriate. All results are in the
quantum bounded error setting.